**3. Homotopic Solution Procedure**

To apply HAM [20], the following initial guesses for Equations (7)–(9), (20), and (21) can be chosen as

$$\begin{array}{l} \psi\_{3,0}(\boldsymbol{\varepsilon}) = \frac{-2\boldsymbol{\varepsilon}^{3}(1+\boldsymbol{\beta}\_{1}+\boldsymbol{\beta}\_{2}) + 3\boldsymbol{\varepsilon}(1+2\boldsymbol{\beta}\_{2})(\boldsymbol{\varepsilon}+2\boldsymbol{\beta}\_{1})}{1+4(\boldsymbol{\beta}\_{1}+\boldsymbol{\beta}\_{2})+12\boldsymbol{\beta}\_{1}\boldsymbol{\beta}\_{2}},\\ \psi\_{1,0}(\boldsymbol{\varepsilon}) = \frac{1-\boldsymbol{\varepsilon}+\boldsymbol{\beta}\_{2}}{1+\boldsymbol{\beta}\_{1}+\boldsymbol{\beta}\_{2}}, \psi\_{2,0}(\boldsymbol{\varepsilon}) = \frac{1-\boldsymbol{\varepsilon}+\boldsymbol{\beta}\_{2}}{1+\boldsymbol{\beta}\_{1}+\boldsymbol{\beta}\_{2}},\\ \psi\_{4,0}(\boldsymbol{\varepsilon}) = \frac{-2\boldsymbol{\beta}^{3}(1+\boldsymbol{\beta}\_{1}+\boldsymbol{\beta}\_{2}) + 3\boldsymbol{\varepsilon}(1+2\boldsymbol{\beta}\_{2})(\boldsymbol{\varepsilon}+2\boldsymbol{\beta}\_{1})}{2(1+4(\boldsymbol{\beta}\_{1}+\boldsymbol{\beta}\_{2})+12\boldsymbol{\beta}\_{1}\boldsymbol{\beta}\_{2})}, \psi\_{5,0}(\boldsymbol{\varepsilon}) = \frac{1-\boldsymbol{\varepsilon}+\boldsymbol{\beta}\_{2}}{1+\boldsymbol{\beta}\_{1}+\boldsymbol{\beta}\_{2}} \end{array} \tag{27}$$

For the initial approximation, the following auxiliary linear operators can be chosen:

$$\begin{array}{l} L(\psi\_3) = \frac{d^4 p}{d\zeta^4}, L(\psi\_1) = \frac{d^2 \theta}{d\zeta^2}, L(\psi\_2) = \frac{d^2 \theta}{d\zeta^2}, \\ L(\psi\_4) = \frac{d^4 \theta}{d\zeta^4}, L(\psi\_5) = \frac{d^2 \theta}{d\zeta^2}. \end{array} \tag{28}$$

which satisfies

$$\begin{aligned} L\_{\psi\_1}[A\_5 + A\_6\varsigma] &= 0, L\_{\psi\_2}[A\_7 + A\_8\varsigma] = 0, \\ L\_{\psi\_3}[A\_1 + A\_2\varsigma + A\_3\varsigma^2 + A\_4\varsigma^4] &= 0, \\ L\_{\psi\_4}[A\_9 + A\_{10}\varsigma + A\_{11}\varsigma^2 + A\_{12}\varsigma^4] &= 0, L\_{\psi\_5}[A\_{13} + A\_{14}\varsigma] = 0. \end{aligned} \tag{29}$$

in which *Ai*(*i* = 1 − 14) are constants of integration.
